Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1332825x+596377125\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1332825xz^2+596377125z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1727341875x+27850481268750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(906, 11103)$ | $3.8901640920921181391335772238$ | $\infty$ |
Integral points
\( \left(906, 11103\right) \), \( \left(906, -12009\right) \)
Invariants
| Conductor: | $N$ | = | \( 95550 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-2403420508125000000$ | = | $-1 \cdot 2^{6} \cdot 3^{6} \cdot 5^{10} \cdot 7^{4} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{11167382937025}{102503232} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-6} \cdot 5^{2} \cdot 7^{2} \cdot 13^{-3} \cdot 2089^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3493768178919323234642699665$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.35954184117841090959518627433$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9827825235301019$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.703572854486686$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.8901640920921181391335772238$ |
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| Real period: | $\Omega$ | ≈ | $0.25938734650543831543633262569$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.0362373652740483884250714610 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.036237365 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.259387 \cdot 3.890164 \cdot 4}{1^2} \\ & \approx 4.036237365\end{aligned}$$
Modular invariants
Modular form 95550.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2643840 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $7$ | $1$ | $IV$ | additive | 1 | 2 | 4 | 0 |
| $13$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 775 & 6 \\ 774 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 623 & 0 \\ 0 & 779 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 391 & 630 \\ 705 & 331 \end{array}\right),\left(\begin{array}{rr} 66 & 565 \\ 325 & 651 \end{array}\right),\left(\begin{array}{rr} 301 & 630 \\ 435 & 331 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[780])$ is a degree-$3622993920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/780\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 15925 = 5^{2} \cdot 7^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $2$ | \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $20$ | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 95550c
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 95550kl1, its twist by $5$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.63700.2 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.210999880000.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.202584375.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.20288450000.8 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.356224786432633190535931125000000000000.7 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.164375807566706672493375000000000000.2 | \(\Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | add | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 1 | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0 | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.